Algebra Theorems Flashcards

Questions and Answers

What does the Remainder Theorem state?

A polynomial f(x) has a factor (x-k) if and only if f(x)=0

Complex imaginary roots must work in teams of two

The remainder of a polynomial divided by x-a is p(a) (correct)

A polynomial of degree n has n roots

What is the Factor Theorem?

A function must have at least one zero

Every rational root of the polynomial can be written in the form of p/q

A polynomial of degree n has n roots

If a polynomial f(x) has a factor (x-k) if and only if f(x)=0 (correct)

What does the Rational Root Theorem indicate?

Every rational root of the polynomial can be written in the form of p/q where p is a factor of the constant term and q is a factor of the leading coefficient.

What does the Irrational Root Theorem state?

If a polynomial has rational coefficients and a + b√c is a root, then a - b√c is also a root. Signup and view all the answers

A polynomial of degree n has n roots.

True Signup and view all the answers

According to the Fundamental Theorem of Algebra, a polynomial function must have at least one zero.

True Signup and view all the answers

What does the Complex Conjugate Theorem state?

Complex imaginary roots must occur in pairs, so if 2-i is a root, then 2+i must also be a root. Signup and view all the answers

What does the Location Principle describe?

If p is a polynomial function and p(x1) and p(x2) have opposite signs, then there is a real number r between x1 and x2 that is a zero of p. Signup and view all the answers

Study Notes

Remainder Theorem

When dividing a polynomial function p(x) by x-a, the remainder r is found by evaluating p(a).

Factor Theorem

A polynomial f(x) contains a factor (x-k) if and only if f(k) equals zero. This signifies k is a root of the polynomial.

Rational Root Theorem

For a polynomial p(x) with integer coefficients, any rational root can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Irrational Root Theorem

In a polynomial p(x) with rational coefficients, if a root takes the form a+b√c (where a and b are rational, and √c is irrational), then its conjugate a-b√c is also a root.

Number of Roots Theorem

A polynomial of degree n possesses exactly n roots, considering both real and complex roots.

Fundamental Theorem of Algebra

Every polynomial function has at least one complex zero, which may be real or non-real.

Complex Conjugate Theorem

If a polynomial contains complex roots, they occur in conjugate pairs; if 2-i is a root, then its conjugate, 2+i, must also be a root.

Location Principle

If a polynomial function p takes on opposite signs at two points, p(x1) and p(x2), then there exists a real number r between x1 and x2 where p(r) equals zero, confirming that r is a root.

Description

Test your knowledge of key algebra theorems with this set of flashcards. Explore concepts such as the remainder theorem, factor theorem, and rational root theorem. Perfect for students wishing to reinforce their understanding of polynomial functions.